Grate the peeled beetroots, slice the onion into thin rings and scrape or grate the horseradish. Set the vegetables aside. Bring the rest of the ingredients for the marinade to the boil in a saucepan, covered, for 15 minutes. Add the beetroots and onion to the saucepan and let them cook for the last two minutes. Remove the saucepan from the heat and add the grated horseradish. Leave the brine to cool to room temperature or place it in the fridge until the next day. Place the herring fillets in a large, sterilised preserving jar, cover them with the cooled brine and refrigerate for at least 3–4 days before serving.
Peel the beetroots and cut them into thick slices. Peel the onion and cut it in half. Place the beetroots and onion in a saucepan with the split peas, cover just with water and simmer, covered, for about 35–45 minutes, until everything is tender. Remove the lid and continue to cook until most of the liquid has evaporated. Take the saucepan off the heat and leave to cool slightly. Then place the beetroot mixture in a food processor or blender, add the oil and blend until smooth. Season to taste with lemon juice, salt and pepper.
Peel the beetroot and slice it as thinly as possible using a mandoline. Place the slices in cold water for a moment, then pat them dry. Whisk the vinegar and oil together and toss the beetroot slices in the mixture. Peel and slice the onion into equally fine rings, then peel, scrape or grate the horseradish. Spread butter on the slices of rye bread, remove the herring fillets from the brine and let them drain briefly. Cut the fillets diagonally so they fit lengthways on the bread slices, and place two pieces of herring on each slice. Mix the beetroot purée with half the onion rings and spread it over the herring. Place a strip of grated horseradish in the centre of each herring sandwich – you may not need it all – and finish with a few onion rings and plenty of dill sprigs.
